Spectral radius and size conditions for 1-tough balanced bipartite graphs to contain a $k$-factor with $k\ge 3$

The largest eigenvalue of the adjacency matrix $A(G)$ of a graph $G$ is called the spectral radius $ρ(G)$ of $G$. If $S$ is a vertex cut of a connected graph $G$, then $c(G-S)$ denotes the number of connected components of $G-S$. The toughness $t(G)$ of a graph $G$ is defined as $t(G) = \min\left\{\frac{|S|}{c(G-S)}: S \subset V(G), \text{ and } c(G-S) > 1\right\}$. A graph \( G \) is \( t \)-tough if \( |S| \ge t c(G - S) \) for every proper subset \( S \subset V(G) \) satisfying \( c(G - S) > 1 \). Combining the toughness and spectral conditions, we provide spectral radius and size conditions for the existence of $k$-factors in 1-tough balanced bipartite graphs. This settles a problem posed by Chen, Fan and Lin in [Discrete Appl. Math. {\bf 355} (2024), 223--231].